Why is there a half (1/2) in front of the formula of finding the area of many shapes?

Area is usually defined in terms of a rectangle (sometimes a square) dating back to the ancient Greeks. Having defined area in terms of a rectangle, then other areas can be found -- particularly the area of a general parallelogram. ( This is often demonstrated by dropping an altitude from one of the vertices, "cutting" off the resulting triangle, and reattaching the triangle on the opposite end of the parallelogram forming a rectangle.)

From the...

Area is usually defined in terms of a rectangle (sometimes a square) dating back to the ancient Greeks. Having defined area in terms of a rectangle, then other areas can be found -- particularly the area of a general parallelogram. ( This is often demonstrated by dropping an altitude from one of the vertices, "cutting" off the resulting triangle, and reattaching the triangle on the opposite end of the parallelogram forming a rectangle.)

From the parallelogram we can get other shapes. Every triangle is 1/2 of a parallelogram. (Flip or reflect the triangle and then rotate until a side matches a side in the original configuration).Likewise, any trapezoid is 1/2 of a parallelogram. (Flip or reflect across the midline, then rotate 180 degrees and match up to original configuration)

Then, many areas are found by cutting a region into triangles --e.g. the formula for the area of a regular polygon is derived by cutting the polygon into congruent triangles getting A=1/2ap.